Step |
Hyp |
Ref |
Expression |
1 |
|
xlebnum.j |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
2 |
|
xlebnum.d |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
3 |
|
xlebnum.c |
⊢ ( 𝜑 → 𝐽 ∈ Comp ) |
4 |
|
xlebnum.s |
⊢ ( 𝜑 → 𝑈 ⊆ 𝐽 ) |
5 |
|
xlebnum.u |
⊢ ( 𝜑 → 𝑋 = ∪ 𝑈 ) |
6 |
|
eqid |
⊢ ( MetOpen ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) = ( MetOpen ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) |
7 |
|
1rp |
⊢ 1 ∈ ℝ+ |
8 |
|
eqid |
⊢ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) = ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) |
9 |
8
|
stdbdmet |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 1 ∈ ℝ+ ) → ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) ) |
10 |
2 7 9
|
sylancl |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) ) |
11 |
|
rpxr |
⊢ ( 1 ∈ ℝ+ → 1 ∈ ℝ* ) |
12 |
7 11
|
mp1i |
⊢ ( 𝜑 → 1 ∈ ℝ* ) |
13 |
|
0lt1 |
⊢ 0 < 1 |
14 |
13
|
a1i |
⊢ ( 𝜑 → 0 < 1 ) |
15 |
8 1
|
stdbdmopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 1 ∈ ℝ* ∧ 0 < 1 ) → 𝐽 = ( MetOpen ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) ) |
16 |
2 12 14 15
|
syl3anc |
⊢ ( 𝜑 → 𝐽 = ( MetOpen ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) ) |
17 |
16 3
|
eqeltrrd |
⊢ ( 𝜑 → ( MetOpen ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) ∈ Comp ) |
18 |
4 16
|
sseqtrd |
⊢ ( 𝜑 → 𝑈 ⊆ ( MetOpen ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) ) |
19 |
6 10 17 18 5
|
lebnum |
⊢ ( 𝜑 → ∃ 𝑟 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 ) |
20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → 𝑟 ∈ ℝ+ ) |
21 |
|
ifcl |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 1 ∈ ℝ+ ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ+ ) |
22 |
20 7 21
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ+ ) |
23 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
24 |
7 11
|
mp1i |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → 1 ∈ ℝ* ) |
25 |
13
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → 0 < 1 ) |
26 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ 𝑋 ) |
27 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ+ ) |
28 |
|
rpxr |
⊢ ( if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ+ → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ* ) |
29 |
27 28
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ* ) |
30 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
31 |
30
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → 𝑟 ∈ ℝ ) |
32 |
|
1re |
⊢ 1 ∈ ℝ |
33 |
|
min2 |
⊢ ( ( 𝑟 ∈ ℝ ∧ 1 ∈ ℝ ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ≤ 1 ) |
34 |
31 32 33
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ≤ 1 ) |
35 |
8
|
stdbdbl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 1 ∈ ℝ* ∧ 0 < 1 ) ∧ ( 𝑥 ∈ 𝑋 ∧ if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ* ∧ if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ≤ 1 ) ) → ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) = ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ) |
36 |
23 24 25 26 29 34 35
|
syl33anc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) = ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ) |
37 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) ) |
38 |
|
metxmet |
⊢ ( ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( Met ‘ 𝑋 ) → ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( ∞Met ‘ 𝑋 ) ) |
39 |
37 38
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( ∞Met ‘ 𝑋 ) ) |
40 |
|
rpxr |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ* ) |
41 |
40
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → 𝑟 ∈ ℝ* ) |
42 |
|
min1 |
⊢ ( ( 𝑟 ∈ ℝ ∧ 1 ∈ ℝ ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ≤ 𝑟 ) |
43 |
31 32 42
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ≤ 𝑟 ) |
44 |
|
ssbl |
⊢ ( ( ( ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ* ∧ 𝑟 ∈ ℝ* ) ∧ if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ≤ 𝑟 ) → ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ) |
45 |
39 26 29 41 43 44
|
syl221anc |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ) |
46 |
36 45
|
eqsstrrd |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ) |
47 |
|
sstr2 |
⊢ ( ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) → ( ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 → ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
48 |
46 47
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 → ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
49 |
48
|
reximdv |
⊢ ( ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑋 ) → ( ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 → ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
50 |
49
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 → ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
51 |
|
oveq2 |
⊢ ( 𝑑 = if ( 𝑟 ≤ 1 , 𝑟 , 1 ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) = ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ) |
52 |
51
|
sseq1d |
⊢ ( 𝑑 = if ( 𝑟 ≤ 1 , 𝑟 , 1 ) → ( ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ↔ ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
53 |
52
|
rexbidv |
⊢ ( 𝑑 = if ( 𝑟 ≤ 1 , 𝑟 , 1 ) → ( ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ↔ ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
54 |
53
|
ralbidv |
⊢ ( 𝑑 = if ( 𝑟 ≤ 1 , 𝑟 , 1 ) → ( ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ↔ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) ) |
55 |
54
|
rspcev |
⊢ ( ( if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ∈ ℝ+ ∧ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) if ( 𝑟 ≤ 1 , 𝑟 , 1 ) ) ⊆ 𝑢 ) → ∃ 𝑑 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ) |
56 |
22 50 55
|
syl6an |
⊢ ( ( 𝜑 ∧ 𝑟 ∈ ℝ+ ) → ( ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 → ∃ 𝑑 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ) ) |
57 |
56
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑟 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ ( 𝑦 ∈ 𝑋 , 𝑧 ∈ 𝑋 ↦ if ( ( 𝑦 𝐷 𝑧 ) ≤ 1 , ( 𝑦 𝐷 𝑧 ) , 1 ) ) ) 𝑟 ) ⊆ 𝑢 → ∃ 𝑑 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ) ) |
58 |
19 57
|
mpd |
⊢ ( 𝜑 → ∃ 𝑑 ∈ ℝ+ ∀ 𝑥 ∈ 𝑋 ∃ 𝑢 ∈ 𝑈 ( 𝑥 ( ball ‘ 𝐷 ) 𝑑 ) ⊆ 𝑢 ) |